Rydberg Formula: Predicting Hydrogen's Spectral Lines

The Rydberg formula predicts the wavelengths of all spectral lines of hydrogen with remarkable accuracy. Discovered empirically by Johannes Rydberg, it was later explained by Bohr's atomic model. One simple constant organizes the entire hydrogen spectrum.

The formula matters because it turned a confusing list of lines into an exact law. It is still used to identify hydrogen in stars and to test atomic theory, and its extensions describe hydrogen-like ions such as He⁺ and Li²⁺.

Key Concepts

Rydberg constant (R_H): For hydrogen, R_H ≈ 1.097 × 10⁷ m⁻¹. It converts level differences into inverse wavelength.

Principal quantum numbers: n₁ is the final (lower) level and n₂ the initial (higher) level, with n₂ > n₁. Integers 1, 2, 3, … select the series.

Hydrogen-like ions: For nuclear charge Z, replace R by Z² R (with reduced-mass corrections). He⁺ (Z = 2) lines are compressed to shorter wavelengths.

Formulas

Rydberg formula: 1/λ = R_H (1/n₁² − 1/n₂²), λ in metres, R_H in m⁻¹. n₁ = 1 gives Lyman, 2 gives Balmer, 3 gives Paschen. Valid for hydrogen; for hydrogen-like ions use 1/λ = Z² R (1/n₁² − 1/n₂²). Assumes vacuum wavelengths and non-relativistic energy levels.

Series limit: When n₂ → ∞, 1/λ_limit = R_H / n₁². For Balmer, λ_limit = 4 / R_H ≈ 365 nm, the shortest Balmer wavelength.

Worked Example

Calculate the wavelength of the second Balmer line (n₂ = 4 → n₁ = 2).

Step 1: 1/λ = 1.097 × 10⁷ × (1/2² − 1/4²) = 1.097 × 10⁷ × (0.25 − 0.0625) = 1.097 × 10⁷ × 0.1875.

Step 2: 1/λ = 2.057 × 10⁶ m⁻¹.

Step 3: λ = 1 / 2.057 × 10⁶ = 4.86 × 10⁻⁷ m = 486 nm.

This matches the observed blue-green H-beta line at 486 nm, confirming the formula to high precision.

Common Misconceptions

• Misconception: The formula works for all atoms unchanged. Correct: It is exact only for hydrogen and hydrogen-like (single-electron) ions with the Z² factor.
• Misconception: n₁ and n₂ can be swapped freely. Correct: n₂ must be larger than n₁ for emission; the term in brackets must be positive.
• Misconception: The Rydberg constant is the same in air measurements. Correct: Wavelengths in air are slightly shorter due to refraction; precision work uses vacuum values.

Real-World Applications

Astronomers use Rydberg predictions to recognize hydrogen and He⁺ lines in stellar spectra and to measure redshifts of distant galaxies. The formula provides wavelength standards for calibrating spectrometers and underpins precision tests of quantum electrodynamics using hydrogen spectroscopy.

In short, 1/λ = R(1/n₁² − 1/n₂²) generates every hydrogen line. Choosing n₁ selects the series; choosing n₂ picks the line within it.